Linear Preservers of Perimeters of Nonnegative Real Matrices

Abstract

Abstract. For a nonnegative real matrix A of rank 1, A can be factored as ab t for somevectors a and b. The perimeter of A is the number of nonzero entries in both a and b. If Bis a matrix of rank k, then B is the sum of k matrices of rank 1. The perimeter of B is theminimum of the sums of perimeters of k matrices of rank 1, where the minimum is takenover all possible rank-1 decompositions of B. In this paper, we obtain characterizations ofthe linear operators which preserve perimeters 2 and k for some k ≥ 4. That is, a linearoperator T preserves perimeters 2 and k(≥ 4) if and only if it has the form T(A) = UAV ,or T(A) = UA t V with some invertible matrices U and V . 1. IntroductionThere is much literature on the study of linear operators that preserve theranks of matrices over several semirings([1]-[8]). Nonnegative matrices also havebeen the subject of research by many authors([2], [5], [6], [8]). Beasley, Gregoryand Pullman [2] obtained characterizations of linear operators which preserve therank of nonnegative real matrices. In [8], Song and Hwang characterized spanningcolumn ranks and their preservers of nonnegative matrices. Beasley, Song, Kangand Sarma [5] treated column ranks of nonnegative real matrices and characterizedtheir preservers.But there are few papers on the characterizations of linear operators preserv-ing the perimeter of matrices. Beasley et al. characterized those linear operatorspreserving the rank and perimeter of Boolean rank-1 matrices([1]).In this paper, we consider the set of linear operators that preserve the perimeterof matrices of rank k(≥ 2) over the nonnegative reals.